On the mass of static metrics with positive cosmological constant -- II

TL;DR

This paper develops a notion of mass for static metrics with positive cosmological constant, establishing bounds for horizons and proving uniqueness of Schwarzschild--de Sitter spacetime.

Contribution

It introduces optimal area bounds for horizons in static metrics with positive cosmological constant and proves a uniqueness theorem for Schwarzschild--de Sitter spacetime.

Findings

Proved positive mass statement characterizing de Sitter as the only static vacuum with zero mass.

Established optimal area bounds for black hole and cosmological horizons.

Deduced a uniqueness result for Schwarzschild--de Sitter spacetime.

Abstract

This is the second of two works, in which we discuss the definition of an appropriate notion of mass for static metrics, in the case where the cosmological constant is positive and the model solutions are compact. In the first part, we have established a positive mass statement, characterising the de Sitter solution as the only static vacuum metric with zero mass. In this second part, we prove optimal area bounds for horizons of black hole type and of cosmological type, corresponding to Riemannian Penrose inequalities and to cosmological area bounds \`a la Boucher-Gibbons-Horowitz, respectively. Building on the related rigidity statements, we also deduce a uniqueness result for the Schwarzschild--de Sitter spacetime.